Fast Algorithms for Refined Parameterized Telescoping in Difference Fields

TL;DR

This paper surveys algorithms for refined parameterized telescoping in difference fields, focusing on efficiency improvements within the framework of $ ext{ extPi} ext{ extSigma}$-extensions for symbolic summation.

Contribution

It introduces new efficient algorithms for refined parameterized telescoping in difference fields, enhancing computational performance in symbolic summation tasks.

Findings

Development of faster algorithms for parameterized telescoping

Improved efficiency in symbolic summation computations

Application to indefinite nested sums and products

Abstract

Parameterized telescoping (including telescoping and creative telescoping) and refined versions of it play a central role in the research area of symbolic summation. Karr introduced 1981 $\Pi\Sigma$-fields, a general class of difference fields, that enables one to consider this problem for indefinite nested sums and products covering as special cases, e.g., the ($q$--)hypergeometric case and their mixed versions. This survey article presents the available algorithms in the framework of $\Pi\Sigma$-extensions and elaborates new results concerning efficiency.